On fictionalism in Aristotle's philosophy of mathematics
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The aim of this dissertation is to show that Aristotle’s ontology cannot provide a model for mathematics. To show this, I argue that (i) mathematical objects must be seen as fictional entities in the light of Aristotle’s metaphysics, and (ii) Aristotle’s mathematical fictionalism is not compatible with his metaphysical realism. My interpretation differs from that of other fictionalists in denying this compatibility. For Aristotle, mathematical objects are “something resulting from abstraction ([Greek phrase]).” For example, geometry investigates a man not qua man, but qua solid or figure. Traditionally, Aristotle’s abstraction has been interpreted as an epistemic process by which a universal concept is obtained from particulars; I rather show his abstraction as a linguistic analysis or conceptual separation by which a certain group of properties are selected: e. g., if a science, X, studies a qua triangle, X studies the properties which belong to a in virtue of a’s being a triangle and ignores a’s other properties. Aristotle’s theory of abstraction implies a mathematical naïve realism, in that mathematical objects are properties of sensible objects. But the difficulty with this mathematical naïve realism is that, since most geometrical objects do not have physical instantiations in the sensible world, things abstracted from sensible objects cannot supply all the necessary objects of mathematics. This is the so-called “precision problem.” In order to solve this problem, Aristotle abandons his mathematical realism and claims that mathematical objects exist in sensibles not as actualities but ‘as matter ([Greek phrase]).’ This claim entails a mathematical fictionalism in metaphysical terms. Most fictionalist interpretations argue that the fictionality of mathematical objects does not harm the truth of mathematics for Aristotle, insofar as objects’ matter is abstracted from sensibles. None of these interpretations, however, is successful in reconciling Aristotle’s mathematical fictionalism with his realism. For Aristotle, sciences are concerned with ‘what is ([Greek phrase])’ and not with ‘what is not ([Greek phrase]).’ Aristotle’s concept of truth rests on a realist correspondence to ‘what is ([Greek phrase])’: “what is true is to say of what is that it is or of what is not that it is not.” Thus, insofar as mathematical objects are fictional, Aristotle’s metaphysics cannot account for the truth of mathematics.